3.16 \(\int (\frac {x^2}{\tan ^{\frac {3}{2}}(a+b x)}-\frac {4 x}{b \sqrt {\tan (a+b x)}}+x^2 \sqrt {\tan (a+b x)}) \, dx\)

Optimal. Leaf size=18 \[ -\frac {2 x^2}{b \sqrt {\tan (a+b x)}} \]

[Out]

-2*x^2/b/tan(b*x+a)^(1/2)

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Rubi [A]  time = 0.12, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {3721} \[ -\frac {2 x^2}{b \sqrt {\tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Tan[a + b*x]^(3/2) - (4*x)/(b*Sqrt[Tan[a + b*x]]) + x^2*Sqrt[Tan[a + b*x]],x]

[Out]

(-2*x^2)/(b*Sqrt[Tan[a + b*x]])

Rule 3721

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^m*(b*Tan[e +
f*x])^(n + 1))/(b*f*(n + 1)), x] + (-Dist[(d*m)/(b*f*(n + 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n + 1)
, x], x] - Dist[1/b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n + 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && LtQ[n
, -1] && GtQ[m, 0]

Rubi steps

\begin {align*} \int \left (\frac {x^2}{\tan ^{\frac {3}{2}}(a+b x)}-\frac {4 x}{b \sqrt {\tan (a+b x)}}+x^2 \sqrt {\tan (a+b x)}\right ) \, dx &=-\frac {4 \int \frac {x}{\sqrt {\tan (a+b x)}} \, dx}{b}+\int \frac {x^2}{\tan ^{\frac {3}{2}}(a+b x)} \, dx+\int x^2 \sqrt {\tan (a+b x)} \, dx\\ &=-\frac {2 x^2}{b \sqrt {\tan (a+b x)}}\\ \end {align*}

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Mathematica [A]  time = 0.96, size = 18, normalized size = 1.00 \[ -\frac {2 x^2}{b \sqrt {\tan (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Tan[a + b*x]^(3/2) - (4*x)/(b*Sqrt[Tan[a + b*x]]) + x^2*Sqrt[Tan[a + b*x]],x]

[Out]

(-2*x^2)/(b*Sqrt[Tan[a + b*x]])

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fricas [A]  time = 0.61, size = 16, normalized size = 0.89 \[ -\frac {2 \, x^{2}}{b \sqrt {\tan \left (b x + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*x/b/tan(b*x+a)^(1/2)+x^2*tan(b*x+a)^(1/2)+x^2/tan(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-2*x^2/(b*sqrt(tan(b*x + a)))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sqrt {\tan \left (b x + a\right )} + \frac {x^{2}}{\tan \left (b x + a\right )^{\frac {3}{2}}} - \frac {4 \, x}{b \sqrt {\tan \left (b x + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*x/b/tan(b*x+a)^(1/2)+x^2*tan(b*x+a)^(1/2)+x^2/tan(b*x+a)^(3/2),x, algorithm="giac")

[Out]

integrate(x^2*sqrt(tan(b*x + a)) + x^2/tan(b*x + a)^(3/2) - 4*x/(b*sqrt(tan(b*x + a))), x)

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maple [F]  time = 0.87, size = 0, normalized size = 0.00 \[ \int -\frac {4 x}{b \sqrt {\tan \left (b x +a \right )}}+x^{2} \left (\sqrt {\tan }\left (b x +a \right )\right )+\frac {x^{2}}{\tan \left (b x +a \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-4*x/b/tan(b*x+a)^(1/2)+x^2*tan(b*x+a)^(1/2)+x^2/tan(b*x+a)^(3/2),x)

[Out]

int(-4*x/b/tan(b*x+a)^(1/2)+x^2*tan(b*x+a)^(1/2)+x^2/tan(b*x+a)^(3/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sqrt {\tan \left (b x + a\right )} + \frac {x^{2}}{\tan \left (b x + a\right )^{\frac {3}{2}}} - \frac {4 \, x}{b \sqrt {\tan \left (b x + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*x/b/tan(b*x+a)^(1/2)+x^2*tan(b*x+a)^(1/2)+x^2/tan(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^2*sqrt(tan(b*x + a)) + x^2/tan(b*x + a)^(3/2) - 4*x/(b*sqrt(tan(b*x + a))), x)

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mupad [B]  time = 3.12, size = 50, normalized size = 2.78 \[ -\frac {x^2\,\sin \left (2\,a+2\,b\,x\right )\,\sqrt {\frac {\sin \left (2\,a+2\,b\,x\right )}{\cos \left (2\,a+2\,b\,x\right )+1}}}{b\,{\sin \left (a+b\,x\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*tan(a + b*x)^(1/2) + x^2/tan(a + b*x)^(3/2) - (4*x)/(b*tan(a + b*x)^(1/2)),x)

[Out]

-(x^2*sin(2*a + 2*b*x)*(sin(2*a + 2*b*x)/(cos(2*a + 2*b*x) + 1))^(1/2))/(b*sin(a + b*x)^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \left (- \frac {4 x}{\sqrt {\tan {\left (a + b x \right )}}}\right )\, dx + \int \frac {b x^{2}}{\tan ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx + \int b x^{2} \sqrt {\tan {\left (a + b x \right )}}\, dx}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*x/b/tan(b*x+a)**(1/2)+x**2*tan(b*x+a)**(1/2)+x**2/tan(b*x+a)**(3/2),x)

[Out]

(Integral(-4*x/sqrt(tan(a + b*x)), x) + Integral(b*x**2/tan(a + b*x)**(3/2), x) + Integral(b*x**2*sqrt(tan(a +
b*x)), x))/b

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